Embibe Experts Solutions for Chapter: Sequence and Series, Exercise 1: Exercise 1

If the average of $20$ different positive integers is $20$ then the greatest possible number among these $20$ numbers can be

The number of terms in an $\mathrm{AP}$ (Arithmetic Progression) is even. The sums of the odd- and even-numbered terms are $24$ and $30$, respectively. If the last term exceeds the first by $10.5$, the number of terms in the $\mathrm{AP}$ is

The maximum sum of the series $20+19\frac{1}{3}+18\frac{2}{3}+18+\dots$ is

If the $nth$ term of an $\mathrm{AP}$ is $\frac{3+n}{4}$, then the sum of first $105$ terms is

If $e^{\left({\cos }^{2}x+{\cos }^{4}x+{\cos }^{6}x+....\infty \right){\log }_{e}2}$ satisfies the equation $t^2-9t+8=0$, then the value of $\frac{2\sin x}{\sin x+\sqrt{3}\cos x}$, where $0<x<\frac{\pi }{2}$, is equal to

If $a, b, c$ form a $G.P.$ with common ratio $r$ such that $0<r<1$, and if $a, \frac{3b}{2}, -4c$ form an $A.P.$, then $r$ is equal to -

In an ordered set of four numbers, the first $3$ are in $\mathrm{AP}$ and the last $3$ are in $\mathrm{GP}$ whose common ratio is $\frac{7}{4}$. If the product of the first and fourth of these number is $49$, then the product of the second and third of these, is

The sum of $\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots +\frac{1}{13\times 14\times 15}+\frac{1}{14\times 15\times 16}$ is $\frac{m}{n}$ in its lowest terms. Find the
value of $n-5 m$